black droppings

I am worried about my new hen. I took in a bird who had last her flock to a dog and was looking for a new home. She is haivng a rough introduction. My girls make her sleep on the door step and wopn't let her on the roost at night. Whan I was cleaning out the coop I noticed that her droppings were very different then the ones under the roost. Black, runny. Is she sick? She's been with me for about a week.

Any new bird you want to introduce should be separated for several weeks, not only for the health of your flock but also for her own. A move, the stress of that and new feed and flockmates....stress expresses itself physiologically in birds and can be even deadly.
I would separate her to a warm temperature stable environment, give her a good general poultry supplement (if she seems to be drinking insufficiently then give an electrolyte in her waterer and put the supplement in her feed)... offer her live culture yogurt free choice .
See how she does on that regime. I would also suggest a faecal float for cocci/worms.

She has been laying an egg a day for the last two days in a row right (6-7 month RIR). I can tell they are hers because my other RIR eggs are twice the size and much darker in color. These are the first eggs she has laid since she got here and I know she had just begun to lay in her previous situation.

Also, I have seen her in the run with the other hens the last two mornings instead of hiding in the coop.

My question is - will it be more stress to pull her now if she is begining to adjust/integrate?